The storage device exploiting the magnetic recording technique or the optical recording technique, or the signal processing apparatus employed in a wireless communication apparatus, as well as the software algorithm, used therefor, is designed on the basis of a linear signal processing theory based on the assumed linear performance of input signals. In general, these input signals are not formed only of perfectly linear signal components, and also contain non-linear components. However, these non-linear components are usually sufficiently low in power and therefore may be approximated to linear signals. For this reason, the signal processing apparatus, based on the linearity theory, has so far been used sufficiently efficaciously.
However, with recent development in the storage technology and increase in the recording density, non-linear properties presented in the reproduced signals have become non-negligible. These non-linear properties account for deterioration in the performance of a phase locked loop (PLL), in the convergence properties of an adaptive equalizing filter or in the ultimate data error rate. Even though attempts have so far been made in increasing the recording density of a recording medium for further improving its recording capacity, non-linear signal distortion, caused by the use of a recording medium having a high recording capacity, or by the use of a detector exhibiting a high detection sensitivity but presenting a non-linear response to input or output signals, proves a significant factor deterrent to further performance improvement of the entire apparatus.
Representative of the causes of these non-linear distortions of the reproduced signals are non-linear properties ascribable to the signal reproducing side and those ascribable to the recording medium. Among the representative causes of the former type, there are a non-linear response and a base line shift of magnetic field-voltage transducing characteristics of an MR (Magneto Resistive) head, used as a reproducing head for magnetic recording, and also a non-linear response of a photodetector used for optical recording. Among the representative causes of the latter type, there are non-linear inter-symbol interference (NLISI) under the state of high recording density of both the magnetic recording medium and the optical recording medium, and vertical signal asymmetry, brought about by the non-linear performance of the reflectivity of the recording medium in the course of optical recording.
The causes for ultimate deterioration in the error rate will now be scrutinized further.
In a linear adaptive equalizing filter, employing an LMS (Least Mean Square) algorithm, as mounted on a customary signal processing apparatus, convergence to tap coefficients, which will give the smallest square error value, is assured by detecting an error signal between a target detection value conforming to a preset equalization system, represented by partial response (PR), and an actually detected signal, as long as an input signal is free of non-linear distortion. However, with the above filter, the non-linear distortion, represented by vertical asymmetry, cannot be corrected, because of its theoretical structure.
It is noted that, with the linear adaptive equalizing filter, in which, due to its algorithm, the tap coefficient which will simply minimize the square error obtained is searched, there unavoidably persists the probability of convergence of the tap coefficients to values different from the ideal values of convergence. This indicates the possibility of producing unforeseen new equalization errors, that is, non-linear equalization errors, due to the fact that, in case an input signal to a linear adaptive equalizing filter suffers from non-linear distortion, resort is had to an adaptive equalizing algorithm in which correction of such non-linear equalization error is intrinsically not presupposed. Such non-linear equalization error leads to deterioration in the ultimate data error rate.
In this consideration, such a technique implementing a polynominal filter as an adaptive equalization filter has been proposed as a method for non-linear equalization of a signal exhibiting non-linear distortion (for example, see Patent Publications 1 and 2). Researches into such polynominal filter, generally termed a Volterra filter, have so far been attempted in a variety of technological fields. With the Volterra filter, it is possible to update the tap coefficients in accordance with an adaptive equalization algorithm, such as LMS or RLS (Recursive Least Square) algorithm, by way of optimization insofar as the least square error is concerned. Detailed explanation on an adaptive equalization Volterra filter may be found in a reference material entitled “Adaptive Polynominal Filters” by V. John Matthews, IEEE SP Magazine, July 1991, pp. 10 to 26.
Meanwhile, a customary second-order Volterra filter is represented by the following equation (1):
                              y          ⁡                      (            k            )                          =                                            ∑                              i                =                0                                                              M                  1                                -                1                                      ⁢                                                            h                                      (                    1                    )                                                  ⁡                                  (                  i                  )                                            ·                              x                ⁡                                  (                                      k                    -                    i                                    )                                                              +                                    ∑                                                i                  1                                =                0                                                              M                  2                                -                1                                      ⁢                                          ∑                                                      i                    2                                    =                  0                                                                      M                    2                                    -                  1                                            ⁢                                                                    h                                          (                      2                      )                                                        ⁡                                      (                                                                  i                        1                                            ,                                              i                        2                                                              )                                                  ·                                  x                  ⁡                                      (                                          k                      -                                              i                        1                                                              )                                                  ·                                                      x                    ⁡                                          (                                              k                        -                                                  i                          2                                                                    )                                                        .                                                                                        (        1        )            
In this equation (1), M1 denotes a tap length of a linear section, and M2 denotes a tap length of a quadratic section. Moreover, in this equation, y(k) denotes an output signal of the second-order Volterra filter, at time k, x(k) is an input signal to the second-order Volterra filter, at time k, h(1)(i) denotes tap coefficients of the linear section, where i=0, 1, . . . , M1−1 and h(2)(i1, i2) denotes tap coefficients of the quadratic section (i1=0, 1, . . . , M2−1; i2=0, 1, . . . , M2−1).
Meanwhile, the second-order Volterra filter can be mounted so that the number of filter taps will sequentially be optimized in accordance with an adaptive equalization algorithm. In addition, if the optimum values of the tap coefficients of the linear and quadratic sections of the second-order Volterra filter are known from the outset, the second-order Volterra filter may also be mounted as a filter of fixed tap coefficients.
For completing calculations of the right side of the equation (1) by one cycle for the input signal x(k), M1 multiplication operations and 2×M2×M2 multiplication operations are needed for the first and second terms of the right side, respectively. Moreover, a number of delay lines for holding the input signals x(k) for the quadratic section, corresponding to M2 clocks, are also needed in addition to the input signal delay line to the linear section of the filter
By exploiting known symmetry of the second-order Volterra filter, the tap coefficients of the quadratic section of the filter satisfy the relationship indicated by the following equation (2):h(2)(i1,i2)=h(2)(i2,i1)  (2)
By exploiting the relationship of this equation (2), the above equation (1) may be simplified to
                              y          ⁡                      (            k            )                          =                                            ∑                              i                =                0                                                              M                  1                                -                1                                      ⁢                                                            h                                      (                    1                    )                                                  ⁡                                  (                  i                  )                                            ·                              x                ⁡                                  (                                      k                    -                    i                                    )                                                              +                                    ∑                              i                =                0                                                              M                  2                                -                1                                      ⁢                                                            h                                      (                    2                    )                                                  ⁡                                  (                                      i                    ,                    i                                    )                                            ·                                                x                  2                                ⁡                                  (                                      k                    -                    i                                    )                                                              +                      2            ⁢                                          ∑                                                      i                    1                                    =                  0                                                                      M                    2                                    -                  1                                            ⁢                                                ∑                                                            i                      2                                        >                                          i                      1                                                                                                  M                      2                                        -                    1                                                  ⁢                                                                            h                                              (                        2                        )                                                              ⁡                                          (                                                                        i                          1                                                ,                                                  i                          2                                                                    )                                                        ·                                      x                    ⁡                                          (                                              k                        -                                                  i                          1                                                                    )                                                        ·                                                            x                      ⁡                                              (                                                  k                          -                                                      i                            2                                                                          )                                                              .                                                                                                          (        3        )            
It is noted that M1 multiplication operations, 2×M2 multiplication operations and M2×(M2−1) multiplication operations are needed for the first, second and third terms of the right side of the equation (3), respectively.
The results of comparison of the number of the multipliers of the quadratic section of the second-order Volterra filter, indicated by the equations (1) and (3), for variable numbers M2, are shown in FIG. 17 and in the following Table 1:
TABLE 1Number of multipliers ofNumber of multipliers ofM2quadratic term of equation (1)quadratic term of equation (2)1222863181243220550306724279856812872916290102001101124213212288156133381821439221015450240
As may be seen from FIG. 17 and Table 1, the larger the value of M2, the more outstanding is the effect of reducing the number of the multipliers by the equation (3). However, even with the configuration of the equation (3), as many as 240 multipliers are needed for the case of M2=15.
That is, even though the higher order Volterra filter is highly efficacious for equalizing an input signal exhibiting non-linear distortion, many multiplication operations are needed if the filter is to be implemented by hardware or software, thus presenting implementation difficulties because of cost.